Question

What is the integral x(2x+1)^0.5 dx?

Answer 1

`\int {x √(2x+1) } \, dx` We will use u-substitution: u=2x +1 , du = 2dx,
`dx= (du)/(2), x= (u-1)/(2)` ...=
`(1)/(4) \int {(u-1)} √(u) \, du = (1)/(4) \int {u^(3/2) -u^(1/2) } \, du`=
=
`(1)/(4) ( (u^(5/2) )/( (5)/(2) )- (u^(3/2) )/( (3)/(2) ) ) =`=
`(1)/(2) (3u^(5/2) -5u^(3/2) )/(15)=`
=
`(1)/(30) u^(3/2) (3u-5)`=
=
`(1)/(30) (2x+1)^(3/2)(6x-2)=`
=
`(1)/(15) (2x+1)^(2/3)(3x-1) +C`